For this type of development to occur, the atmosphere established in the classroom ought to assure that everyone's estimate is important and valued, that students feel comfortable taking risks, and that explanation and justification of estimation strategies is a regular part of the process. Estimations of measurements as well as of quantities should pervade the classroom activity. As students communicate with each other about how their estimates are formulated, they further develop their personal bank of strategies for estimation.

Activities which provide experiences for the student to determine reasonableness of answers and to establish the difference between estimated answers and exact answers as well as when the use of each is appropriate, should be developed through non-routine problem solving activities that involve measurement, quantities, and computation.

In the high school grades, estimation and number sense are much more important skills than algorithmic pencil-and-paper computation. Students should become masters at applying estimation strategies so that answers displayed on a calculator can be instinctively compared to a sense of the range in which the correct answer lies. With calculators and computers being used on a consistent basis in these grade levels, it is critical that students understand the displays that occur on the screen and the effects of calculator rounding either because of the calculators own operational system or because of user-defined constraints. Issues of number of significant figures and what kinds of answers make sense in a given problem setting create new reasons to a focus on reasonableness of answers.

Measurement settings are rich with opportunities to develop an understanding that estimates are often used to determine approximate values which are then used in computations and that results so obtained are not exact but fall within a range of tolerance. Issues also appropriate for discussion at this level include acceptable limits of tolerance, and assessments of the degree of error of any particular measurement or computation.

Another topic appropriate at these grade levels is the estimation of probabilities and of statistical phenomena like measures of central tendency or variance. When statisticians talk about "eyeballing" the data, they are explicitly referring to the process where these kinds of measures are estimated from a set of data. The skill to be able to do that is partly the result of knowledge of the measures themselves and partly the result of years of experience in computing them. Students can begin to develop these skills as well.

Students should, by this point in their educations, understand that sometimes an estimate will be an accurate enough number to serve as an answer. At other times, an exact computation will need to be done, either mentally, with paper and pencil, or with a calculator to arrive at a more precise answer. Which procedure should be used is dependent on the setting and the problem. Even in cases where exact answers are to be calculated, however, students must understand that it is almost always a good idea to have an estimate in mind before the actual exact computation is done so that the computed answer can be checked against the estimated one.

Building upon K-8 expectations, experiences in grades 9-12 will be such that all students:

M. determine the reasonableness of answers to problems solved using pencil-and-paper techniques, mental math, algebraic formulas and equations, computers or calculators.

• Students are constantly asked if the answers theyve computed make sense. Latisha's calculator displayed 17.5 after she entered 3 times the square root of 5. Is this a reasonable answer?
• Students are sometimes presented with hypothetical scenarios that challenge both their estimation and technology skills: During a test, Paul entered

  	y = .516x - 2 	and 	y = .536x + 5
  

  in his graphics calculator . After analyzing the two lines displayed on the "standard" screen window setting [-10,10, -10,10], he decided to indicate that the lines were parallel and that there was no point of intersection. Was Paul's answer reasonable?

• Another example: Jim used the zoom feature of his graphics calculator and found the solution to y = 2x + 3 and y = -2x - 1 to be (-.9997062,1.0005875). When he got his paper back his teacher had taken points off. What was wrong with Jim's response?

• Students use tables of data from an almanac to make estimates of the means and medians of a variety of measures from average state population to average percentage of voters in presidential elections. Any table where a list of figures, but no mean, is given can be used for this kind of activity. After estimates are given, actual means and medians can be computed and compared to the estimates. Reasons for large differences between the means and medians ought to also be explored.
• Students collect data about themselves and their families for a statistics unit on standard deviation. After everyone has entered data in a large class chart regarding number of siblings, distance lived away from school, oldest sibling, and many other pieces of numerical data, the students work in groups to first estimate and then compute means and medians as a first step toward a discussion of variation.
• Students each track the performance of a particular local athlete over a period of a few weeks and use whatever knowledge they have about past performance to predict his or her performance over the next week. They provide as detailed and statistical a prediction as possible. At the end of the week, predictions are compared to the actual performance.

• Students in the high school grades learn methods for estimating the magnitude of error in their estimations at the same time as they learn the main procedures. Discussions regarding the acceptability of a given magnitude of error are a regular part of classroom activity when estimation is being used.
• Students work in small groups to measure the linear dimensions of a rectangular box and determine its volume using measures to the nearest 1/8 inch since that is the smallest unit on their rulers. After their best measurements and computation, the groups share their estimates of the volume and discuss differences. Each group then constructs a range in which they are sure the actual exact answer lies by using a measure for each dimension which is clearly short of the actual measure and multiplying them and finding a measure for each dimension which is clearly longer than the actual measure and multiplying them. The range is specified as between those two products.
• Students are presented with these two solutions to the following problem and discuss the error associated with each approach:

  How many kernels of popcorn are in a cubic foot of popcorn?

  1. There are between 3 and 4 kernels of popcorn in 1 cubic inch. There are 1728 cubic inches in a cubic foot. Therefore there are 6048 kernels of popcorn in a cubic foot. [3.5 (the average number of kernels in a cubic inch) x 1728 = 6048]
  2. The diameter of a kernel of popcorn is approximately 9/16". The volume of this "sphere" is 0.09314 cubic inches. Therefore (1728/0.09314) = 18552.71634 or 18,000 pieces of popcorn.

• Students act as Quality Assurance officers for mythical companies and devise procedures to keep errors within acceptable ranges. One possible scenario:

  In order to control the quality of their product, Paco's Perfect Potato Chip Company guarantees that there will never be more than 1 burned potato chip for every thousand that are produced. The company packages the potato chips in bags that hold about 333 chips. Each hour 9 bags are randomly taken from the production line and checked for burnt chips. If more than 15 burnt chips are found within a four hour shift, steps are taken to reduce the number of burnt chips in each batch of chips produced. Will this plan ensure the company's guarantee?

• Students regularly review statistical claims reported in the media as to their accuracy given the data that is available. One possible case study (the idea for which was taken from: Landwehr, J. Exploring Surveys and Information from Samples. Palo Alto CA: Dale Seymour, 1987, page 2.):

  The March, 1985 Gallup survey asked 1,571 American adults "Do you approve or disapprove of the way Ronald Reagan is handling his job as president."

  56% said that they approved. For results based on samples of this size, one can say with 95% confidence that the error attributable to sampling and other random effects could be as much as 3 percentage points in either direction.

  A newspaper editor read the Gallup survey report and created the following headline:
  BARELY ONE-HALF OF AMERICA APPROVES OF THE JOB REAGAN IS DOING AS PRESIDENT.

  Did this editor make appropriate use of the data for the headline?